Perturbative approach to continuous-time quantum error correction

We present a discussion of the continuous-time quantum error correction introduced by J. P. Paz and W. H. Zurek [Proc. R. Soc. A 454, 355 (1998)1364-502110.1098/rspa.1998.0165]. We study the general Lindbladian which describes the effects of both noise and error correction in the weak-noise (or strong-correction) regime through a perturbative expansion. We use this tool to derive quantitative aspects of the continuous-time dynamics both in general and through two illustrative examples: the three-qubit and five-qubit stabilizer codes, which can be independently solved by analytical and numerical methods and then used as benchmarks for the perturbative approach. The perturbatively accessible time frame features a short initial transient in which error correction is ineffective, followed by a slow decay of the information content consistent with the known facts about discrete-time error correction in the limit of fast operations. This behavior is explained in the two case studies through a geometric description of the continuous transformation of the state space induced by the combined action of noise and error correction.